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arxiv: 2311.03112 · v1 · submitted 2023-11-06 · ✦ hep-th

Recursive construction for expansions of tree Yang-Mills amplitudes from soft theorem

Chang Hu , Kang Zhou This is my paper

Pith reviewed 2026-05-24 06:03 UTC · model grok-4.3

classification ✦ hep-th
keywords tree-level amplitudessoft theoremsYang-Mills amplitudesgauge invariancerecursive constructionamplitude expansionsBCJ numerators
0
0 comments X

The pith

A recursive procedure expands tree-level Yang-Mills amplitudes into scalar amplitudes using only the soft behavior of external gluons while preserving explicit gauge invariance at every step.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a bottom-up recursive method to express Yang-Mills amplitudes in terms of Yang-Mills-scalar and bi-adjoint scalar amplitudes. The construction begins with the known soft limits of gluons and adds terms step by step until the full amplitude is recovered. Gauge invariance is maintained explicitly throughout the recursion, so that an invariant starting expression produces an invariant final result. This yields expansions that can be combined with existing results for Einstein-Yang-Mills amplitudes to produce gauge-invariant BCJ numerators. A reader would care because the method relies solely on intrinsic soft behavior rather than additional computational structures.

Core claim

A fundamentally different bottom-up approach expands tree-level Yang-Mills amplitudes into Yang-Mills-scalar amplitudes and bi-adjoint scalar amplitudes. The method relies solely on the intrinsic soft behavior of external gluons. The recursive procedure consistently preserves explicit gauge invariance at every step, ultimately resulting in a manifest gauge-invariant outcome when the initial expression is already framed in a gauge-invariant manner. The resulting expansion can be directly analogized to expansions of gravitational amplitudes. When combined with expansions of Einstein-Yang-Mills amplitudes, it yields gauge-invariant BCJ numerators.

What carries the argument

The recursive construction driven by the soft theorem for external gluons, which builds the amplitude expansion term by term while enforcing gauge invariance at each stage.

If this is right

  • The expansions remain manifestly gauge invariant whenever the input expression is gauge invariant.
  • The method produces expansions directly analogous to those obtained for gravitational amplitudes via double-copy relations.
  • Combining the Yang-Mills expansions with existing Einstein-Yang-Mills expansions produces gauge-invariant BCJ numerators.
  • The procedure applies to any tree-level amplitude whose soft limits are known.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same recursion might generate expansions for amplitudes with more external legs once the soft factors are supplied.
  • Similar soft-driven recursions could be tested on other gauge theories whose soft theorems are known.
  • If the uniqueness holds, the method offers an independent check on amplitude expressions derived from different starting points.

Load-bearing premise

The soft behavior of external gluons by itself is sufficient to uniquely determine the full expansion without any further input.

What would settle it

An explicit low-point amplitude whose soft-consistent expansion obtained by the recursion differs from the known result computed by other means.

Figures

Figures reproduced from arXiv: 2311.03112 by Chang Hu, Kang Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1: Two 5-point diagrams. Figure (a1) and (a2) are the same tree diagram, while (a1) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Turn the 4-point vertex to 3-point ones. The bold line corresponds to the inserted prop [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

In this paper, we have introduced a fundamentally different approach, based on a bottom-up methodology, to expand tree-level Yang-Mills (YM) amplitudes into Yang-Mills-scalar (YMS) amplitudes and Bi-adjoint-scalar (BAS) amplitudes. Our method relies solely on the intrinsic soft behavior of external gluons, eliminating the need for external aids such as Feynman rules or CHY rules. The recursive procedure consistently preserves explicit gauge invariance at every step, ultimately resulting in a manifest gauge-invariant outcome when the initial expression is already framed in a gauge-invariant manner. The resulting expansion can be directly analogized to the expansions of gravitational (GR) amplitudes using the double copy structure. When combined with the expansions of Einstein-Yang-Mills amplitudes obtained using the covariant double copy method from existing literature, the expansions presented in this note yield gauge-invariant BCJ numerators.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript introduces a bottom-up recursive construction to expand tree-level Yang-Mills amplitudes into linear combinations of Yang-Mills-scalar (YMS) and bi-adjoint scalar (BAS) amplitudes. The procedure is asserted to rely solely on the known soft theorem for external gluons, to preserve explicit gauge invariance at each recursive step, and—when combined with existing covariant double-copy expansions of Einstein-Yang-Mills amplitudes—to produce gauge-invariant BCJ numerators.

Significance. If the recursion uniquely fixes all coefficients from soft limits alone, the result would supply a new, rule-independent route to amplitude expansions and BCJ numerators that complements existing double-copy literature. The explicit preservation of gauge invariance throughout the construction would be a useful technical feature.

major comments (2)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: The central assertion that 'the intrinsic soft behavior of external gluons alone is sufficient to uniquely determine the full expansion into YMS and BAS amplitudes without any additional input' is load-bearing. Soft theorems fix only the leading singular terms; the manuscript must show explicitly (via an example recursion or inductive argument) how the finite parts and the precise linear combination within the YMS/BAS basis are fixed without implicit extra conditions such as basis choices or vanishing requirements at finite kinematics.
  2. The manuscript supplies no worked examples, explicit coefficient tables, or low-point checks that would allow verification that the recursion produces the known expansions (e.g., the four-gluon or five-gluon cases). Without such concrete output, the claim that the procedure works and is unique cannot be assessed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for identifying areas where the manuscript's claims require stronger explicit support. We agree that concrete demonstrations are needed to substantiate the uniqueness of the recursion from soft theorems alone. We will revise the manuscript to include an inductive argument and low-point examples, addressing both major comments directly.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] The central assertion that 'the intrinsic soft behavior of external gluons alone is sufficient to uniquely determine the full expansion into YMS and BAS amplitudes without any additional input' is load-bearing. Soft theorems fix only the leading singular terms; the manuscript must show explicitly (via an example recursion or inductive argument) how the finite parts and the precise linear combination within the YMS/BAS basis are fixed without implicit extra conditions such as basis choices or vanishing requirements at finite kinematics.

    Authors: We accept that the abstract claim requires explicit justification beyond the current presentation. The recursion is constructed so that each step applies the soft theorem to determine the leading pole, after which gauge invariance (preserved by construction) constrains the finite remainder to a unique linear combination in the YMS/BAS basis. In the revision we will add a short inductive argument, starting from the three-point seed and showing at each order how the soft limit plus the requirement that the result remain gauge invariant fixes all coefficients without extra kinematic assumptions. revision: yes

  2. Referee: The manuscript supplies no worked examples, explicit coefficient tables, or low-point checks that would allow verification that the recursion produces the known expansions (e.g., the four-gluon or five-gluon cases). Without such concrete output, the claim that the procedure works and is unique cannot be assessed.

    Authors: We agree that the absence of explicit low-point verifications makes the claims difficult to assess. The revised manuscript will contain a new section with fully worked four-gluon and five-gluon expansions, including the coefficient tables that reproduce the known YMS/BAS decompositions. These examples will also serve as the base cases for the inductive argument mentioned above. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation grounded in external soft theorem

full rationale

The provided abstract and description present a recursive construction explicitly based on the known soft behavior of gluons, with the claim that this suffices without Feynman/CHY input and that gauge invariance is preserved step-by-step. No quoted equations or sections show a self-definitional reduction, a fitted parameter renamed as prediction, or a load-bearing self-citation chain that collapses the result to its own inputs. Dependence on external double-copy literature is cited as combination rather than internal justification. The derivation chain therefore remains self-contained against the stated external benchmark (soft theorem).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based on abstract only; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The method rests on the standard soft theorem of Yang-Mills theory.

axioms (1)
  • domain assumption The soft behavior of external gluons is sufficient to determine the full recursive expansion into YMS and BAS amplitudes
    Abstract states the method relies solely on the intrinsic soft behavior of external gluons.

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discussion (0)

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Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Hidden zeros for higher-derivative YM and GR amplitudes at tree-level

    hep-th 2025-10 unverdicted novelty 6.0

    Hidden zeros extend to higher-derivative tree-level gluon and graviton amplitudes, with systematic cancellation of propagator singularities shown via bi-adjoint scalar expansions.

  2. On soft factors and transmutation operators

    hep-th 2024-06 unverdicted novelty 6.0

    Reconstruction of known soft factors via transmutation operators and proof of nonexistence of higher-order universal soft factors for YM and GR amplitudes.

  3. Constructing tree amplitudes of scalar EFT from double soft theorem

    hep-th 2024-06 unverdicted novelty 6.0

    A method constructs tree amplitudes of scalar EFTs from the double soft theorem by determining the explicit double soft factor during the construction process.

  4. Multi-trace YMS amplitudes from soft behavior

    hep-th 2024-01 unverdicted novelty 6.0

    Derives expansion formulas for multi-trace YMS amplitudes bottom-up from soft gluon and scalar behaviors.

  5. Can Locality, Unitarity, and Hidden Zeros Completely Determine Tree-Level Amplitudes?

    hep-th 2026-04 unverdicted novelty 5.0

    Locality, unitarity, and hidden zeros determine tree-level YM and NLSM amplitudes by reconstructing their soft theorems.

  6. $2$-split from Feynman diagrams and Expansions

    hep-th 2025-08 unverdicted novelty 5.0

    Proof via Feynman diagrams that tree-level BAS⊕X amplitudes with X=YM,NLSM,GR obey 2-split under kinematic conditions, extended to pure X amplitudes with byproduct universal expansions of X currents into BAS currents.

  7. Soft theorems of tree-level ${\rm Tr}(\phi^3)$, YM and NLSM amplitudes from $2$-splits

    hep-th 2025-05 unverdicted novelty 5.0

    Extends a 2-split factorization approach to reproduce known leading and sub-leading soft theorems for Tr(φ³) and YM single-soft and NLSM double-soft amplitudes while deriving higher-order universal forms and a kinemat...

  8. Note on hidden zeros and expansions of tree-level amplitudes

    hep-th 2025-02 unverdicted novelty 4.0

    Hidden zeros in tree-level amplitudes of several theories are attributed to zeros of bi-adjoint scalar amplitudes via universal expansions, with a mechanism shown to cancel potential propagator divergences in gravity.

  9. Towards tree Yang-Mills and Yang-Mills-scalar amplitudes with higher-derivative interactions

    hep-th 2024-06 unverdicted novelty 4.0

    Extends soft-behavior approach to construct tree YM and YMS amplitudes with F^3 (and F^3+F^4) insertions as universal expansions, plus a conjectured general formula for higher-mass-dimension YM amplitudes from ordinary ones.

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